Fr. Robert Barron's video refers to the following:
John G. Messerly's article "Religion's smart-people problem: The shaky intellectual foundations of absolute faith" (Salon, 2014/12/21); and
Charts from "What Do Philosophers Believe?" (pdf) by David Bourget and David J. Chalmers (2013/11/30), which give very basic information about the degree to which 1,972 professional philosophers agreed on "thirty central philosophical issues" when asked in a survey, and how those answers correlated to their specializations.
Surprisingly, Fr. Barron is not convinced by Messerly's argument. Here are his points (I'll flesh this out later):
Apparently (to paraphrase Barry Goldwater), intellectual dishonesty in defense of theism is no vice.